Abstract: The classical Yamabe problem—solved through the work of Yamabe, Trudinger, Aubin, and Schoen—asserts that on any closed smooth connected Riemannian manifold $(M^n,g)$, $n\geq 3$, one can find a metric conformal to $g$ with constant scalar curvature. A fully nonlinear analogue replaces the scalar curvature by symmetric functions of the Schouten tensor. Traditionally, the existing theory has required the scalar curvature to have a fixed sign. In a recent work, we broaden the scope of fully nonlinear Yamabe problem by establishing optimal Liouville-type theorems, local gradient estimates, and new existence and compactness results. Our results allow conformal metrics with scalar curvature of varying signs. A crucial new ingredient in our proofs is our enhanced understanding of solution behavior near isolated singularities. I will also discuss extensions to manifolds with boundary, treating prescribed boundary mean curvature and the boundary curvature arising from the Chern–Gauss–Bonnet formula.

Joint with Analysis seminar at 3pm.

In the 1950’s, E. Calabi  initiated a far-reaching program of finding, on a given compact Kähler manifold $X$, a canonical representative of the space of Kähler metrics that belong to a fixed de Rham class. He proposed as a candidate of such representative the notion of constant scalar curvature Kähler metric, including the Kähler–Einstein metrics  as a special case.  Calabi’s program was one of the most active areas of research in Kähler geometry during the last half-century. The central conjecture in the field, which is still open in full generality, is the Yau–Tian–Donaldson (YTD) conjecture. It states, broadly speaking, that the full obstruction for the existence of a constant scalar curvature Kahler metric can be expressed in terms of a complex-algebraic notion of K-polystability of  $X$. This correspondence, if established, will have further deep implications for the definition of well-behaved moduli spaces of Kähler manifolds. 

In the 1990’s, an extension of Kähler geometry emerged from studies in $(2,2)$ supersymmetric quantum field theory in physics. These geometric structures were later rediscovered, and given the name of generalized Kähler (GK) structures, in the context of Hitchin’s generalized geometry program. In the ensuing decades it has become clear that GK geometry is a deeply structured extension of Kähler geometry with novel implications for complex, symplectic and Poisson geometry.

In this talk I will explain how,  guided by an infinite dimensional momentum map picture,  one can extend Calabi’s notion of constant scalar curvature Kahler metric to the  generalized Kahler context. This setup will naturally lead us to an algebro-geometric notion of Poisson K-polystability of a polarized complex Poisson manifold,  and to a Yau-Tian-Donaldson type conjecture on such manifolds. I will discuss a resolution of this conjecture on the complex projective space.  

This talk is based on joint works with Jeffrey Streets, Yury Ustinovskiy and Brent Pym.

The smooth mean curvature flow often develops singularities, making weak solutions essential for extending the flow beyond singular times, as well as having applications for geometry and topology. Among various weak formulations, the level set flow method is notable for ensuring long-time existence and uniqueness. However, this comes at the cost of potential fattening, which reflects genuine non-uniqueness of the flow after singular times. Even for flows starting from smooth, embedded, closed initial data, such non-uniqueness can occur. Thus, we can't expect genuine uniqueness in general. Addressing this non-uniqueness issue is a difficult problem. With Alec Payne, we establish an intersection principle comparing two intersecting flows. We prove that level set flows satisfy this principle in the absence of non-uniqueness.

Weighted-cscK metrics provide a universal framework for the study of canonical metrics, e.g., extremal metrics, Kahler-Ricci soliton metrics, \mu-cscK metrics. In joint works with Yaxiong Liu, we prove that on a Kahler manifold X, the G-coercivity of weighted Mabuchi functional implies the existence of weighted-cscK metrics.  In particular, there exists a weighted-cscK metric if X is a projective manifold that is weighted K-stable for models.  We will also discuss some progress on singular varieties.

Abstract: The geometric flow of hypersurfaces is an interesting and active area. Its importance lies in the applications in geometry and topology. For example, Huisken and Ilmanen in 2001 applied the inverse mean curvature flow to prove the famous Penrose conjecture. Brendle-Guan-Li proposed a conjecture on the Alexandrov-Fenchel inequalities for hypersurfaces in the sphere and introduced a locally constrained fully nonlinear curvature flow to study this conjecture.  In this talk, we will discuss using a new type of flow to study this question and some recent progress on this conjecture.